M1 June 2014 Q1

Figure 1: particle C hanging from strings AC and BC inclined at 30 and 50 degrees to a horizontal ceiling — Figure 1

A particle of weight \(W\) newtons is attached at \(C\) to two light inextensible strings \(AC\) and \(BC\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(AC\) and \(BC\) inclined to the horizontal at 30\(^\circ\) and 50\(^\circ\) respectively, as shown in Figure 1.

Given that the tension in \(BC\) is 6 N, find

(a) the tension in \(AC\), (3)

(b) the value of \(W\). (3)

Mark scheme (a)

Scheme	Marks
Resolving horizontally: \(T\cos 30^\circ = 6\cos 50^\circ\)	M1A1
\(T = 4.45\) (N), 4.5 (N), or better	A1
	(3)

Notes

First M1 for resolving horizontally with correct no. of terms and both \(T_{AC}\) and ‘6’ terms resolved.

First A1 for a correct equation in \(T_{AC}\) only.

Second A1 for 4.5 (N), 4.45 (N) or better. (4.453363194)

N.B. The M1 is for a complete method to find the tension so where two resolution equations, neither horizontal, are used, the usual criteria for an M mark must be applied to both equations and the first A1 is for a correct equation in \(T_{AC}\) only (i.e. \(W\) eliminated correctly)

Alternatives:

Triangle of Forces : \(\dfrac{T_{AC}}{\sin 40^\circ} = \dfrac{6}{\sin 60^\circ}\) (same equation as → resolution) M1A1

Lami’s Theorem: \(\dfrac{T_{AC}}{\sin 140^\circ} = \dfrac{6}{\sin 120^\circ}\) (same equation as → resolution) M1A1

Mark scheme (b)

Scheme	Marks
Resolving vertically: \(W = 6\cos 40^\circ + T\cos 60^\circ\)	M1A1
\(= 6.82\) (N), 6.8 (N), or better	A1
	(3)
	(6 marks)

Notes

First M1 for resolving vertically with correct no. of terms and both \(T_{AC}\) (does not need to be substituted) and ‘6’ terms resolved.

First A1 for a correct equation in \(T_{AC}\) and \(W\).

Second A1 for 6.8 (N), 6.82 (N) or better. (6.822948256)

Alternatives:

Triangle of Forces : \(\dfrac{6}{\sin 60^\circ} = \dfrac{W}{\sin 80^\circ}\) M1A1

Or Lami’s Theorem: \(\dfrac{6}{\sin 120^\circ} = \dfrac{W}{\sin 100^\circ}\) M1A1

Or Resolution in another direction e.g. along one of the strings M1 (usual criteria) A1 for a correct equation.